224 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



153. Defining the instantaneous parabola of a projectile in a medium 

 whose resistance is proportional to the square of the velocity as that which 

 would be described if the resistance ceased to act, prove that its latus rectum 

 diminishes at a rate which varies as v 3 cos 2 6, where 6 is the inclination to the 

 horizon of the direction of motion at the point where the velocity is v. Prove 

 also that the axis of the parabola moves towards or from the point of 

 projection according as the projectile is ascending or descending. 



154. Show that the horizontal and vertical coordinates x, y of a particle 

 moving under gravity in a medium whose resistance is R satisfy the equation 



v being the velocity and &amp;lt; the inclination of the tangent to the horizontal. 



155. Prove that the time t, the horizontal abscissa x, and the vertical 

 ordinate y, at a point where the tangent of the inclination of the velocity to 

 the horizon is p, of a trajectory in a medium whose resistance varies as the 

 nth power of the velocity, are given by 



fa i-d 



= I (1 +p 2 ) 2 dp, 

 J P 



where 



w denoting the terminal velocity in the medium, and a the tangent of the 

 inclination to the horizon at the origin, the point of infinite velocity. 



156. A particle in a medium whose resistance is small, and equal to 

 ic (velocity) 2 , is executing small vibrations. Prove that the period is unaltered, 

 but that in each semi -vibration the amplitude is diminished by ^/ca 2 , where a 

 is the original amplitude. 



157. A pendulum oscillates in a medium of which the resistance per unit 

 of mass is &amp;lt; (velocity) 2 . Prove that, when powers of the arc above the first 

 are neglected, the period is the same as in the absence of resistance, but the 

 time of descent exceeds that of ascent by %Ka*J(fi/g}, where a is the angular 

 amplitude of the descent, and I is the length of the pendulum. 



158. Prove that in a resisting medium a particle can describe a circle 

 of diameter a under the action of a force to a point on the circumference 

 varying inversely as the fourth power of the distance, the resistance being 

 proportional to r~*J(a 2 -r 2 ) when the distance is r. 



159. A particle describes an equiangular spiral in a resisting medium 

 under a force F to the pole, and the rate of description of areas is uniformly 



retarded ; prove that 



F=p.r-*-\r~\ 



where X and p. are constants, and find the law of resistance. 



